with(combinat);

pet_autom2cycles :=
proc(src, aut)
local numa, numsubs;
local marks, pos, cycs, cpos, clen;

    numsubs := [seq(src[k]=k, k=1..nops(src))];
    numa := subs(numsubs, aut);

    marks := Array([seq(true, pos=1..nops(aut))]);

    cycs := []; pos := 1;

    while pos <= nops(aut) do
        if marks[pos] then
            clen := 0; cpos := pos;

            while marks[cpos] do
                marks[cpos] := false;
                cpos := numa[cpos];
                clen := clen+1;
            od;

            cycs := [op(cycs), clen];
        fi;

        pos := pos+1;
    od;

    return mul(a[cycs[k]], k=1..nops(cycs));
end;

pet_cube_symact :=
proc(cv, symp, symf)
local bincbverts, bin, f;
    bincbverts :=
    [[0,0,0], [0,0,1], [0,1,0], [0,1,1],
     [1,0,0], [1,0,1], [1,1,0], [1,1,1]];


    bin := bincbverts[cv];
    bin := [seq(bin[symp[q]], q=1..3)];

    for f to 3 do
        if f in symf then
            bin[f] := 1 - bin[f];
        fi;
    od;

    1 + bin[1]*4 + bin[2]*2 + bin[3];
end;


pet_cycleind_tcube_verts :=
proc()
local bincbverts, cbedg, sl,
    perm, flip, cind, verts, vperm;

    cbedg :=
    [{1,2}, {1,3}, {1,5},
     {2,4}, {2,6}, {3,4},
     {3,7}, {4,8}, {5,6},
     {5,7}, {6,8}, {7,8}];

    verts :=
    [seq(seq([edg, v], v in edg),
         edg in cbedg)];

    cind := 0;

    for perm in permute(3) do
        for flip in powerset(3) do
            sl :=
            [seq(q=pet_cube_symact(q, perm, flip),
                 q=1..8)];

            vperm := subs(sl, verts);

            cind := cind +
            pet_autom2cycles(verts, vperm);
        od;
    od;

    cind/48;
end;


pet_cycleind_tcube_faces :=
proc()
local perm, flip, cind, sl, faces, fperm;
    cind := 0;

    faces :=
    [1,2,3,4,5,6,7,8,
     {1,2,3,4}, {1,2,5,6},
     {2,4,6,8}, {1,3,5,7},
     {3,4,7,8}, {5,6,7,8}];

    for perm in permute(3) do
        for flip in powerset(3) do
            sl :=
            [seq(q=pet_cube_symact(q, perm, flip),
                 q=1..8)];

            fperm := subs(sl, faces);

            cind := cind +
            pet_autom2cycles(faces, fperm);
        od;
    od;

    cind/48;
end;

Xverts :=
subs([seq(a[q]=n, q=1..24)], pet_cycleind_tcube_verts());

Xfaces :=
subs([seq(a[q]=n, q=1..14)], pet_cycleind_tcube_faces());

pet_cycleind_tcube_edges :=
proc()
local bincbverts, cbedg, sl,
    perm, flip, cind, edges, eperm, e1, e2, f1, f2;

    cbedg :=
    [{1,2}, {1,3}, {1,5},
     {2,4}, {2,6}, {3,4},
     {3,7}, {4,8}, {5,6},
     {5,7}, {6,8}, {7,8}];

    edges := [seq(e, e in cbedg)];

    for e1 to nops(cbedg) do
        for e2 from e1+1 to nops(cbedg) do
            f1 := cbedg[e1]; f2 := cbedg[e2];

            if nops(f1 intersect f2) = 1 then
                edges :=
                [op(edges), {f1, f2}];
            fi;
        od;
    od;

    cind := 0;

    for perm in permute(3) do
        for flip in powerset(3) do
            sl :=
            [seq(q=pet_cube_symact(q, perm, flip),
                 q=1..8)];

            eperm := subs(sl, edges);

            cind := cind +
            pet_autom2cycles(edges, eperm);
        od;
    od;

    cind/48;
end;

Xedges :=
subs([seq(a[q]=n, q=1..36)], pet_cycleind_tcube_edges());

A274900 := unapply(Xverts, n);
A274901 := unapply(Xfaces, n);
A274902 := unapply(Xedges, n);